Differential models of production change in the marginal cost and the multi product firm
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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm. Lecture XXVI. Change in the Marginal Cost. Shares of Marginal Cost Since both total and marginal cost depend on output levels and input prices, we start by considering marginal share of each input price.

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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

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Differential models of production change in the marginal cost and the multi product firm

Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

Lecture XXVI


Change in the marginal cost

Change in the Marginal Cost

  • Shares of Marginal Cost

    • Since both total and marginal cost depend on output levels and input prices, we start by considering marginal share of each input price


Differential models of production change in the marginal cost and the multi product firm

  • Based on this definition, we define a Firsch price index for inputs as


Differential models of production change in the marginal cost and the multi product firm

  • Completing the single output model


Multiproduct firm

Multiproduct Firm

  • Expanding the production function to a multiproduct technology


Differential models of production change in the marginal cost and the multi product firm

  • Expanding the preceding proof

    • Computing the first-order conditions


Differential models of production change in the marginal cost and the multi product firm

  • Now we replicate some of the steps from the preceding lecture, allowing for multiple outputs.

    • Taking the differential of the first-order condition with respect to each output


Differential models of production change in the marginal cost and the multi product firm

  • Again note by the first-order condition

  • Thus


Differential models of production change in the marginal cost and the multi product firm

  • With


Differential models of production change in the marginal cost and the multi product firm

  • Differentiating with respect to the input prices yields the same result as before


Differential models of production change in the marginal cost and the multi product firm

  • Slightly changing the preceding derivation by differentiating the production function by a vector of output levels, holding prices and other outputs constant yields


Differential models of production change in the marginal cost and the multi product firm

  • Multiplying through by γyields

  • Using the tired first-order conditions


Differential models of production change in the marginal cost and the multi product firm

  • With


Differential models of production change in the marginal cost and the multi product firm

  • Differentiating the production function with respect to yields


Differential models of production change in the marginal cost and the multi product firm

  • Collecting these equations:

    • Differentiating the first-order conditions with respect to ln(z’)

    • Differentiating the first-order conditions with respect to ln(p’)


Differential models of production change in the marginal cost and the multi product firm

  • Differentiating the production function with respect to ln(z’)

  • Differentiating the production function with respect to ln(p’)


Differential models of production change in the marginal cost and the multi product firm

  • The extended form of the differential supply system is then.

    • Starting with the total derivative of ln(q)

    • Premultiplying by F


Differential models of production change in the marginal cost and the multi product firm

  • Note by the results from Barten’s fundamental matrix


Differential models of production change in the marginal cost and the multi product firm

  • θir is the share of the ith input in the marginal cost of the rth product.

  • Summing this marginal cost over all inputs


Differential models of production change in the marginal cost and the multi product firm

  • Defining the matrix


Introduction of quasi fixed variables

Introduction of Quasi-Fixed Variables

  • Expanding the differential model further, we introduce quasi-fixed variables into the production set


Differential models of production change in the marginal cost and the multi product firm

  • Following Livanis and Moss, the differential supply function for this specification becomes


Differential models of production change in the marginal cost and the multi product firm

  • Starting with the input demand system, we add a random disturbance relying on the theory of rational random behavior (RRB, Theil 1975):


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